Hodge Theory and Moduli

Contemporary Mathematics Volume 766, 2021 https://doi.org/10.1090/conm/766/15381

Hodge theory and moduli

Phillip Griﬃths

Outline I.A Introductory comments I.B Classiﬁcation problem; ﬁrst case of relation between moduli and Hodge theory II.A Moduli II.B Hodge theory III.A Generalities on Hodge theory and moduli III.B I-surfaces and their period mappings Appendix; Application of Hodge theory to I-surfaces with a degree 2 elliptic singularity

I.A. Introductory comments. Algebraic geometry is frequently seen as a very interesting and beautiful subject, but one that is also very difficult to get into. This is partly due to its breadth, as traditionally algebra (commutative and homological), topology, analysis, differential geometry, Lie theory — and more recently combinatorics, logic, categorical and derived algebraic techniques, . . . — are used to study it. I have tried to make these notes accessible to a general audience by reviewing some of the most basic concepts, illustrating the material with elementary examples and informal geometric and heuristic arguments, and with occasional side comments for experts in the subject. Although algebra, both commutative and homological, are the central tools in algebraic geometry, their use in many current areas of research (birational geometry and the minimal model program) is frequently quite technical and will not be extensively discussed in these notes. On the other hand, partly because Hodge theory involves analysis, Lie theory and differential geometry as well as a wide variety of homological methods, it is perhaps less prevalent in the more algebraic works in the field. One objective of these talks is to illustrate how, in partnership with the more algebraic and homological methods, Hodge theory may be used to study interesting and important geometric questions. In the appendix we have sketched how this may be carried out in a particular example.

Notes based on the lecture presented at the conference “Geometry at the frontier” held at Pucon, Chile during November 2018. Parts of this paper represents joint work in progress with Mark Green, Radu Laza and Colleen Robles.

c 2021 American Mathematical Society 163

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In summary the theme of these notes is to discuss and illustrate how complex analysis, diﬀerential geometry and Lie theory may be combined to study a basic problem in algebraic geometry. Although the main techniques in much of contemporary algebraic geometry are algebraic, some of the most interesting questions in the subject require non-algebraic methods for their study and this is what we hope to illustrate.1 As a warm up and to establish some notations we recall that an aﬃne algebraic variety is given by the solution space over the complex numbers to polynomial equations

(∗) fi(x1,...,xn)=0 i =1,...,m. The “elementary” examples are

¨¨ linear spaces ax + by + c =0 ¨

conics ax2 +2bxy + cy2 + ex + fy + g =0 n n quadrics Q(x)= aijxixj + bixi + c =0,aij,aji i,j=1 i=1 The ﬁrst non-elementary examples are cubics

y2 =4x3 + ax + b

The first two elementary examples and the non-elementary example are algebraic curves. One of course considers higher dimensional varieties; surfaces, threefolds,. . . We will be primarily concerned with curves and surfaces. The above algebraic curves are all affine algebraic varieties in C2. In general one adds to an affine variety the asymptotes or “points at infinity” to obtain the n n n−1 n n+1 projective space P = C ∪ P∞ .Equivalently,P is the quotient of C \{0} ∗ n by the scaling action zi → λzi where λ ∈ C . Geometrically P is the set of lines through the origin in Cn+1. The picture of the projective plane P2 is

line at inﬁnity

1This paper is written in an informal style, usually without precise deﬁnitions and statements of results. At the end there is a fairly extensive set of references where the more standard presen- tation of the material to algebraic geometers may be found. In particular we suggest [CMSP17] as a general reference relating Hodge theory and algebraic geometry and where several examples relevant to these notes are discussed in detail.

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and the picture in P2 of the hyperbola in C2 given by y x2 − y2 =1 x

is something like

The equations of the completion in Pn of an aﬃne variety given in Cn by (∗) are obtained by homogenizing: set xi = zi/z0 and clear denominators to obtain

fi(z0,z1,...,zn)=0

di where fi(λz0,λz1,...,λzn)=λ fi(z0,zi,...,zn), di =degfi. Then the hyperbola above becomes 2 − 2 2 z1 z2 = z0 .

Later on we will consider varieties in weighted projective spaces P(a0,a1,...,an) ∗ ai n+1 ∗ where λ ∈ C acts on zi by λ(zi)=λ zi, and in the quotient C \{0}/C the algebraic varieties are deﬁned by weighted homogeneous polynomials. Here the weights ai are positive integers and gcd(a1,...,aN ) = 1. The following will be used in our discussion of the I-surfaces: Example: P(1, 1, 2) is embedded in P3 by → 2 2 [x0,x1,y] [x0,x0x1,x1,y]. The image is

2 z0z2 = z1 , which is

Two algebraic varieties are considered to be equivalent if there is a “change of variables” that transforms one into the other. Thus if the discriminant b2 − ac =0 all conics are equivalent to the circle 2 2 2 z1 + z2 = z0 . Initially changes of variables were linear transformations (including projections); later on rational changes of variables became allowed. Historically algebraic varieties arose from two sources: projective geometry (lines and linear spaces, conics and higher dimensional quadrics), and from complex analysis. In complex analysis the issue was to understand the integrals r(x, y(x))dx, f(x, y(x)) = 0

of algebraic functions, and the “functions” deﬁned by inverting the integrals x(w) w = r(x, y(x))dx.

Here f(x, y) is a polynomial and r(x, y) is a rational function; y(x) is a multi- valued “algebraic function” deﬁned by f(x, y(x)) = 0. The integral depends on

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choosing a path γ of integration in the complex plane and a branch y(x)off(x, y(x)) = 0 along γ.2 Thus sin w dx sin w dx w = √ = 1 − x2 y(x) where x2 + y2 =1and dx2 + dy2 = dx/y gives the parametrization w → (sin w, cos w) of the circle by arclength in terms of “elementary” functions (trigonometric functions and logarithms). However parametrizing the ellipse by arclength led to integrals such as p(w) dx w = √ 4x3 + ax + b which gave non-elementary functions and led Euler, Legendre, Abel, Gauss, Jacobi, Riemann,. . . to the beginnings of the rich and deep interplay between analysis and algebraic geometry. This evolved into modern Hodge theory, and it is this interface between analysis and algebraic geometry that is a main theme of these talks. Part of the richness of the subject of algebraic geometry are the multiple perspectives that may be used in its study: • geometric; • algebraic — e.g., as we will brieﬂy discuss, in birational geometry the algebraic classiﬁcation of certain classes of singularities

of algebraic varieties plays a central role;3 • analytic; we have mentioned complex analysis and the integrals of algebraic functions; • topological; one always has in mind the ﬁrst picture

that is encountered. Example (running example for illustrative purposes): The compact 1- dimensional complex manifold associated to the algebraic curve 2 X = {y =(x − a1)(x − a2) ···(x − a2g+1)}

where the ai are distinct is a compact Riemann surface of genus g which can be studied from all of these perspectives.

2A more correct notation would be w = (x(w),y(w)) r(x, y(x))dx where (x(w),y(w)) is the point on the curve f(x, y)=0inC2 that is given by the integral along the path γ in the curve. To simplify the notation the second coordinate y(w) will be implicit in what follows. 3Cf. [Kol13b] in which the extent and complexity of this story are on full display.

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For g = 1 as in the cubic above, in addition to the topological one the pictures are

algebraic

+ ∞ a1 a2 a3 analytic

- ∞

where a1,a2,a3 are the roots of the cubic equation in the RHS of the above equation. The + and − represent the two possible values of y(x) with the un- derstanding that analytic construction across a slit changes the sign of y(x)= (x − a1)(x − a2) ···(x − a2g+1). The integral w dx y is determined up to the periods dx dx π1 = ,π2 = ; δ y γ y

one may show that π1 =0andIm( π2/π1) > 0. Incidentally for the hyperbola y2 = x2 − 1 the picture is

- ; dx there is only one cycle δ. In this case there is only one period δ y . For another analytic perspective, as will be further discussed below the function p(w) given by inverting the elliptic integral is a doubly periodic entire function 4 (doubly due to the periods δ dx/y and γ dx/y) that leads to the parametrization of the cubic curve C /X ∈ ∈ w /(p(w),p(w))

4For the hyperbola above there is only one period and inverting the integral gives singly periodic trigonometric functions.

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We note that y(w)=p(w) arises from p(w) dx p(w)dw dw = d = . y y(w)

Here the ratio λ = π2/π1 of the periods is determined up to aλ + b λ → cλ + d ab ∈ Z ∈ Z with cd SL2( ) reflecting the choices of the basis δ, γ H1(X, )with(δ, γ)=1. I.B. Classification problem; first case of relation between moduli and Hodge theory. A central problem in algebraic geometry is to classify the equivalence classes of algebraic varieties. For this there are two types of parameters:

• discrete (e.g., the genus g =(1/2)b1(X) for smooth algebraic curves) • continuous (moduli; the smooth curves of genus g form a (3g−3+ρ)-dimensional family Mg,whereρ =dimAut(X)). Example (curves): g = 0 there is only one equivalence class; for example using the birational map given by stereographic projection all non-singular conics are projectively equivalent and are birationally equivalent to a line P1

(x(t),y(t)) t

where x(t),y(t) are rational functions of t.

g =1 dimM1 = 1, from the theory of elliptic curves on has that 1728a3 j = , Δ=a3 − 27b2 Δ gives a set-theoretically 1-1 map M → C; thus the curves 1 3 3 3 P2 x0 + x1 + x2 = x0x1x1 (in ) 3 Q1(x0,x1,x2,x3)=Q2(x0,x1,x2,x3)=0 (inP ) are both equivalent to y2 =4x3 + ax + b for a unique value of j. Here Q1(x)=0andQ2(x) = 0 are smooth quadrics that intersect transitively.5 In general one hopes that

5We are ﬁnessing the subtle issues that arise when the curve has non-trivial automorphisms. We refer to [Kol13a] and the references given there for a discussion of the structure of moduli spaces at points where the corresponding algebraic variety has automorphisms.

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(i) a moduli space M will be an algebraic variety, generally not complete (compact); (ii) there will be a canonical completion M corresponding to adding certain singular varieties. In these notes we will assume (i) and will be primarily concerned with (ii). What does (ii) mean? We imagine a family of plane curves

Xt = {f(x, y, t)=0},t∈ Δ that are smooth for t ∈ Δ∗ but may be singular for t = 0. A picture like

t 0

t = 0 t = 0 will give such a family.6 Applying coordinate changes depending on t can give a family Xt such that Xt is equivalent to Xt for t = 0 but X0 is quite diﬀerent from X0. For example we can have

and even

How can we say what a canonical choice for X0 should be? Historically one suggested answer to this question was provided by Hodge theory; i.e., considering the period matrix associated to the curve. For the example y2 = x(x − t)(x − 1)

0 t 1

as t turns around 0 by following what γ does we get a picture like the ﬁgure on the left

∼

and denoting homology by ∼ the picture on the right leads to the result that as t turns around the origin in homology we have the Picard-Lefschetz transformation δ → δ γ → γ + δ.

6Below we will give equations for such a picture.

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• π1(t) is non-zero and holomorphic for t ∈ Δ; • log t ∈ π2(t)=π1(t) 2πi + (holomorphic function of t Δ). ∗ In general for any family Xt, t ∈ Δ , of smooth genus 1 curves we will have periods π1,π2 as above where

λ = π2/π1, Im λ>0;

here we are thinking of λ as a point in SL2(Z)\H where H = {w ∈ C :Imw>0} is the upper half plane. The periods are locally holomorphic functions of t ∈ Δ∗,and as t turns around the origin the cycles δ, γ will undergo a monodromy transformation ( π2 ) → ab ( π2 ) , where π1 cd π1 ab ∈ Z T = cd SL2( ) is the monodromy matrix. For w a lifting of λ to H this gives a diagram z ∈ H w /H

2πiz ∗ π / e = t ∈ Δ SL2(Z)\H where π(t)=λ and w(z +1)=Tw(z). Lemma 1. The eigenvalues μ of T satisfy |μ| =1. Since the characteristic polynomial of T has integral coeﬃcients, by the Gelfand- Schneider theorem from analytic number theory μ = e2πip/q is root of unity. Replacing t by tq gives that T is unipotent, and for a suitable choice of generators of H (X , Z) we may assume that 1 t 1 m T = ,m∈ Z+. 01 Thus any monodromy matrix is equivalent to a power of a Picard-Lefschetz transformation. Lemma 2. Given a holomorphic mapping w : H → H satisfying w(z +1)=w(z)+m, it follows that w(z)=mz + u(e2πiz) where u is bounded as Im z →∞. Taking m = 1 for simplicity this gives log t π(t)= + u(t), 2πi where u(t) is holomorphic in all of Δ. Both of these lemmas are proved by complex analysis arguments using the Schwarz lemma in the form

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w is distance decreasing in the SL2(R)-invariant hyperbolic (Poincar´e) dw dw metric |w|2 and the observation that the length of the circles |t| = tendstozeroas → 0.

Sketch of the proof of Lemma 1: Let zn ∈ H be a sequence with Re zn =0, Im zn →∞and set wn = w(zn). Then

w(zn +1)=Tw(zn)=Twn.

The hyperbolic distance d(zn,zn +1) → 0, and by the distance decreasing property of w we have 7 d(wn,Twn) → 0.

Now wn = An · w for some An ∈ SL2(R) and ﬁxed w ∈ H, and using the invariance of the metric from −1 → d(w, An TAnw) 0 a little argument shows that by passing to a subsequence we will have −1 → { } An TAn H = isotropy group of w . Since H is compact, all its eigenvalues have absolute value 1 and this implies the same for T .8 Since all that we have used is that the ratio of the periods λ(t)of any holomorphic family of genus 1 curve parametrized by Δ∗ has the same analytic behavior as for the above family has a pair of roots of the cubic coming together at t =0wemaydrawthe Conclusion: The periods of an arbitrary family of genus 1 algebraic curves over Δ∗ have the same asymptotic behavior as a family acquiring nodal singularities given locally analytically by x2 = y2 + tf(x, y). The local pictures are

analytic local picture of

topological local picture of

This analysis of the asymptotics of the period matrix (Hodge structure) extends to that of algebraic curves of any genus g 2 and provided an early suggestion as to what Mg should be.

7 The horizontal segments from zn to zn + 1 map down to circles in the punctured disc. 8 R H R Since SL( ) acts transitively on , the isotropy group of any point is conjugate in SL2( ) { cos θ sin θ } ∈ H to the isotropy group − sin θ cos θ of i .

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The schematic of M2 is

This gives the stratification of M2 together with the incidence (degeneration) relations among the strata. (The solid and dotted arrows will be explained later.) We will see that this stratification is captured by the Hodge structures and their limits. The objective of these notes is to discuss how this picture might be extended to some completed moduli spaces of varieties of general type (analogues of curves of genus g 2) and to illustrate how this works for the first non-classical algebraic surfaces (called I-surfaces and which have the invariants pg(X)=2,q(X)=0, 2 KX =1). To jump ahead and anticipate some of the main points to be made; with notations and terminology to be explained, there are first the general results (some of which are work in progress). • For a given class of surfaces of general type the moduli space M exists and has a canonical completion M. • There is a period mapping

M −→Φ P ⊂ Γ\D

that associates to each surface X theHodgestructureonH2(X, Z). • There is a canonical minimal completion P of P to projective variety, and the period mapping extends to M −Φ−→e P.

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There are then the speciﬁc results for the I-surface X. We will foreshadow these here with the explanation of the notations and terminology to also be given later. The objective is to give something of the ﬂavor of what is to come. We begin with the

Picture/equations: ⎧ ⎪ ⎪ ⎪ P → P3 → 2 2 ⎨⎪P (1, 1, 2) given by (t0,t1) [t0,t0t1,t1,y] • V ⎪ ⎪ ⎪X → P(1, 1, 2)is a 2:1 map branched over the vertex P and the inter- ⎩⎪ section of P(1, 1, 2) with a quintic surface V ∈|OP3 (5)|; • X is realized as the weighted hypersurface in P(1, 1, 2, 5) given by an equa- 2 9 tion z = F10(t0,t1,y).

For the moduli space, MI is smooth and

1 10 • dim MI = h (TX ) = 28; • the period domain DI is a homogeneous contact manifold with dim DI = 57=2dimMI +1; • Φ=MI → ΓI \DI and Φ∗ is injective (local Torelli)

⇓

Φ(MI ) = contact submanifold P → ΓI \DI .

The contact structure on P will be explained below. In the appendix by analyzing the Hodge structure at the boundary of moduli we will sketch a proof of generic local Torelli. On the Hodge theoretic side we have the

Picture of the stratiﬁcation of P (N = logarithm of monodromy):

9This is the bicanonical model of a smooth I-surface. 10 0 2 In fact, it can be shown that for a smooth X, h (TX )=h (TX ) = 0, and from the Hirzebruch Riemann-Roch theorem the Euler characteristic χ(TX )=−28. One may also “count parameters” in the above equation for X and arrive at the same conclusion.

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2 0 N = 0 1 2 I 1 N = 0, rank N = 2

1 2 2 N = 0, N = 0, rank N = 1 II III rank N = 4

1 1 N 2 = 0, rank N=3 and IV rank N2 = 1

2 N 2 = 0, 2 V rank N = 2

The main result, here stated informally, is The Hodge theoretic stratiﬁcation of P uniquely determines the stratiﬁcation MGor 11 of I . Rather than display the whole picture, the following table is just the part for simple elliptic singularities (types I and III). They have N 2 =0sinceforthesemi- stable-reduction (SSR) of such a degeneration only double curves (and no triple points) occur; all of the other types occur if we include cusp singularities.

k minimal − codim stratum dimension (9 di) k resolution X in MI i=1

I0 28 canonical singularities 0 0 0

I2 20 blow up of a K3-surface 7 1 8 I1 19 minimal elliptic surface with χ(X)=2 8 1 9

III2,2 12 rational surface 14 2 16

III1,2 11 rational surface 15 2 17

III1,1,R 10 rational surface 16 2 18

III1,1,E 10 blow up of an Enriques surface 16 2 18 III1,1,2 2ruledsurfacewithχ(X)=0 23 3 26 III1,1,1 1ruledsurfacewithχ(X)=0 24 3 27

The particular K3, elliptic surfaces, rational surfaces and ruled surfaces can be speciﬁed. Note that the last column is the sum of the two columns preceding it.

11 MGor Here I is the completed moduli space for Gorenstein I-surfaces. Work in progress suggests that the Hodge theoretic structure of P may go a long way towards determining the full stratiﬁcation of MI by singularity type.

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(This may be explained using Hodge theory; in the appendix we will sketch how this may be done for I2.) II.C. Moduli and Hodge theory. We have seen that Hodge theory, in the classical form of periods of integrals of algebraic functions together with some complex analysis and differential geometry, suggests what singular curves should be included to compactify the moduli space Mg, leading to an essentially smooth Mg. For surfaces (and higher dimensional) varieties of general type the story thus far is both similar and different,12 especially in the non-classical (term to be explained) case. Birational geometry tells us that it is possible to define a moduli space M with a canonical completion M.Itdoesnot (i) tell us what the singular surfaces X corresponding to the boundary ∂M = M\M are;13 (ii) tell us the stratification of M;and (iii) in contrast to the curve case, M may be highly singular along M\M,and it does not suggest how to desingularize it. We will explain and illustrate how Hodge theory, in partnership with birational geometry, helps us understand points (i)–(iii). A. Moduli Invariants of a smooth projective variety X are basically Kodaira dimension κ(X) ¨¨ • discrete HH topological (Chern numbers); and • continuous (moduli). 0 m X is a compact, complex manifold and a basic invariant is the space H (KX )of global holomorphic forms expressed locally in holomorphic coordinates x1,...,xn as m ϕ = f(x)(dx1 ∧···∧dxn) where f(x) is holomorphic and transforms by the mth power of the Jacobian deter- minant when we change coordinates. The Kodaira dimension κ(X) is defined by 0 0 κ(X) dim H (mKX )=h (mKX )=Cm + ··· ,C >0. 0 By convention we set κ(X)=−∞ if all h (mKX )=0. The purpose of this part of these notes is to give an informal introduction to moduli, to describe two simple classes of algebraic curves and surfaces, and to illustrate the semi-log-canonical (slc) singularities that arise for surfaces and to begin the discussion of how Hodge theory relates to them. Examples: When n =1andX is the smooth algebraic curve (compact Riemann surface) with affine equation 2g+1 2 y = (x − ai),ai distinct, i=1

12Cf. the papers [Kol13a], [Kol13b] and the recent S´eminaire Bourbaki by Benoist in the references. 13In fact, even though the non-Gorenstein isolated singularities are all rational, there does not yet seem to be a practical bound on their index.

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we may picture X as a 2-sheeted branched covering of P1

ai ∞ and the holomorphic 1-forms on X are p(x)dx ϕ = , deg p(x) g − 1. y q(x)dxm − − For m 2 there are similar expressions y where deg q(x) (2g 2)m g for the holomorphic m-forms. The genus g(X)=g and the number of parameters of X’s given as above is 2g +1− 2=2g − 1; a general curve of genus of genus g is represented this way 0 0 1 for g =1, 2. The ϕ’s above give the space H (KX )=H (ΩX ) of holomorphic differentials on X, and the first de Rham cohomology group is a direct sum 1 ∼ 0 1 ⊕ 0 1 HDR(X) = H (ΩX ) H (ΩX ) ∼ which using de Rham’s theorem gives the Hodge structure on H1(X, C) = H1 (X, C). Thus for h0(K )=dimH0(K )wehave DR X X 1 h0(K )= b (X)=g, X 2 1 0 the first result in Hodge theory relating the algebro-geometric invariant h (KX )to the topological invariant b1(X). Now take X to be the smooth algebraic surface with affine equation (∗) z2 = f(x, y), deg f(x, y)=2k where f(x, y) = 0 defines a smooth algebraic curve C ⊂ P2.14 The holomorphic 2-forms on X are p(x, y)dx ∧ dy ϕ = , deg p(x, y) k − 3.15 z 0 ThereareformulassimilartotheaboveinthecurvecasefortheH (mKX )’s. 0 2 0 For H (ΩX )=H (KX ) the direct sum decomposition ∗∗ 2 ∼ 0 2 ⊕ 1 1 ⊕ 0 2 1 1 1 ( ) HDR(X) = H (ΩX ) H (ΩX ) H (ΩX ),H(ΩX )=H (ΩX ) 2 C ∼ 2 16 gives the Hodge structure on H (X, ) = HDR(X). 1 1 For an initial explanation of the H (ΩX )term,ifQ is the bilinear form on H2(X, C) given by the cup-product, then 2 2 C 0 2 F H (X, ) := H (ΩX ) ∩ 1 2 C 0 2 ⊕ 1 1 F H (X, ) := H (ΩX ) H (ΩX ) where under the cup product Q in cohomology F 1 = F 2⊥ 14This surface is similar to but both simpler and more complicated than the I-surface. 15Here p(x, y) is arbitrary. Later in these notes we will encounter the case k = 3, i.e., X is a P2 0 2 2-sheeted covering branched over a sextic curve in :thisisaK3surfacewithh (ΩX )=1. 16In contrast to the curve case, for any k 4 a general surface in the moduli space M of surfaces of the above numerical type is equivalent to one given by (∗).

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and 1 ∩ 1 1 1 F F = H (ΩX ) ∗∗ 0 2 so that the Hodge decomposition ( ) is determined by H (ΩX )andQ.Thusin 0 n both these cases, H (ΩX ) together with Q determines the Hodge decomposition and resulting Hodge structure on the cohomology Hn(X, C), n =dimX. The Kodaira number κ(X) for curves is ⎧ ⎨⎪−∞ for g(X)=0 κ(X)= 0for g(X)=1 ⎩⎪ 1for g(X) 2. For the above algebraic surfaces ⎧ ⎨⎪−∞ k 2 (rational) κ(X)= 0 k = 3 (K3) ⎩⎪ 2 k 4(generaltype) to get κ(X) = 1 you have to allow C to be quite singular. General type surfaces are those with κ(X) = 2; for these the important numerical invariants are • 0 0 2 pg = h (KX )=h (ΩX ) = geometric genus; • 0 1 q = h (ΩX ) = irregularity; • 2 2 KX = c1(X) . They are related by • − 1 2 pg q +1= 12 (KX + χtop(X)) (Noether’s formula); K2 • X 17 pg 2 + 2 (Noether’s inequality). Theorem ([KSB88], [Ale94]). For general type surfaces with given numerical invariants there exists a moduli space M with a canonical completion M. As noted above the proof is via birational geometry. It describes in principle what the singularities of a surface X corresponding to a boundary point in M\M can be.18 For surfaces there is no description, nor examples that I know other than the work of [FPR15a,FPR15b,FPR17], of the global structure of what the X’s can be and how they fit together. Some guiding questions are • How does Hodge theory limit what the singular X’s can be? • Which Hodge-theoretically possible degenerations are realized algebro- geometrically? • Does the Hodge theoretic stratification capture the algebro-geometric one? ∗ π ∗ −1 Given a family X −→ Δ of smooth surfaces Xt = π (t)fort =0,bythe theorem there is defined a unique limit surface X0 = X that fills in the family X → Δ where the conditions

17 2 For the above surface we have KX =2andpg = 3, so that it is extremal for Noether’s 2 inequality. For the I-surface we have KX =1andpg = 2 so that it is also extremal. 18In the paper [Kol13a] by Koll´ar in the references there is a fairly short list of the singularity types that can occur. However within each non-Gorenstein type there is an invariant, the index. The recent paper [RU19] seems to be a promising approach to bounding the index in terms of 2 KX .

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19 • X has canonical singularities over Xsing; and • X is of relative general type and minimal (more precisely, the relative dualizing sheaf ωX/Δ is Q-Cartier and relatively ample). The ﬁrst condition is local along X; the second is global. As is the case for any analytic variety, M has a stratiﬁcation

• M is a union of irreducible subvarieties Zi; • the incidence relation Zj ⊂ Zi means that singular varieties parametrized by Zi can degenerate further into those parametrized by Zj . The proof of the general existence theorem does not suggest what the stratification should be; again aside from [FPR15a, FPR15b, FPR17]Iknowofno other examples where it has been analyzed. To give some flavor of how Hodge theory helps to organize the singularities, we note that X∗ → Δ∗ is topologically a fibre bundle over the circle, and thus there is ∗ a monodromy operator (here t0 ∈ Δ is a base point) 2 Z → 2 Z 20 T : H (Xt0 , ) H (Xt0 , ).

Denoting by T = TssTu

the Jordan decomposition of T where Tss is semi-simple and Tu is unipotent with logarithm N, using analytic arguments arising from Hodge theory that extend the one given above in the case of elliptic curves leads to a proof the monodromy theorem [PS08], [Sch73] m Tss = 1 (i.e., the eigenvalues of T are roots of unity) N 3 = 0 (i.e., the Jordan blocks of T have length 2). A crude Hodge theoretic classification of the singularities of X is given by • normal (a): N =0 • normal (b): N =0, N 2 =0 • normal (c): N 2 =0 • non-normal (a): N = 0 but N 2 =021 • non-normal (b): N 2 =0. This may be refined by putting in the ranks of N and of N 2. A much finer invariant is given by also including the conjugacy class of Tss, usually expressed in terms of the spectrum. And if we include the extension data in the limiting mixed Hodge structure (LMHS), we obtain even more Hodge-theoretic information.22 The following is an informal discussion of typical singularities of each of the above types.

19 0 For normal X this means that for U any open set in X any holomorphic ω ∈ H (U∩Xreg,KX) satisfies ω ∧ ω<∞ Xreg . 20We will consider integral cohomology modulo torsion. 21In the examples I know of, non-normal =⇒ N =0. 22The above crude Hodge theoretic classification is extracted from its associated graded. The LMHS will be defined below.

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Normal (a): Then the monodromy is ﬁnite and the Hodge structures on the 2 23 H (Xt, C) extend across t =0. These include a number of ﬁnite quotient singularities; typically among them are those denoted 1 (1,a), gcd (d, a)=1 d and which is given by the quotient of C2 acted on by the cyclic group generated by (x, y) → (ζx,ζay) 2πi/d 1 − where ζ = e . Among these are the Wahl singularities n2 (1,na 1). For n =2 this is a cone over a rational normal curve C in P4. It is noteworthy in that for this singularity T =Id. Normal (b): simple elliptic singularities. Here (X, p) is a normal surface X having an isolated singular point p, the minimal resolution (X,C ) → (X, p)isthen given by contracting an elliptic curve C ⊂ X with C2 = −d where d>0isthe degree of the elliptic singularity. To say that (X, C)isminimalmeansthatthere are no (−1)-curves not meeting C.24 The assumption that (X, p) is smoothable implies that 1 d 9. For d 3, one may think of the cone over an elliptic normal curve in Pd−1. One typically pictures such a singularity as C

There are two types of restriction here: (i) the cone is over a smooth elliptic curve as opposed to a cone over a curve of genus g 2; (ii) the restriction d 9 for the elliptic curve. An analytic explanation may be given for (i); as noted above, (ii) is the condition that the isolated singularity be smoothable. Note: We are finessing the subtlety that in order to fit the desingularization X of X into a family X → Δ we have to do semi-stable-reduction (SSR), which involves ˜m m X → a base change t = t where Tss = Id. The fibre over the origin in ΔhasX as one component. The other component is a rational surface Y meeting X along 0 2 25 C,andp (Y ) = 0 so that lim → H (Ω ) lives on X. g t 0 Xt For the normal (b) degeneration a similar argument applies except now for 0 2 0 2 ωt ∈ H (Ω ) and limt→0 ωt := ω ∈ H (Ω ) Xt X ∈ 0 1 ∼ C ResC (ω) H (ΩC ) = .

23There is a Riemann extension theorem for a family of Hodge structures over Δ∗ having ﬁnite monodromy. Thus even though X0 may be singular, there is associated to it a polarized Hodge structure. ∼ 24These are curves E = P1 with E2 = −1. 25An additional subtlety is that in order to have some degree of uniqueness one may want to allow X to have cyclic quotient singularities of a “simpler” type than the ones that were started with. The meaning of lim → H0(Ω2 ) will be discussed below. t 0 Xt

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If there are e elliptic singularities, this type of argument leads to the bound

e rank N.

Another type of Hodge theoretic argument gives

e rank N +1.

We will see that both bounds are sharp for I-surfaces. We will also see that for I-surfaces the degrees of the elliptic singularities are determined by the eigenvalues of Tss. An explicit such singular surface will be discussed in the appendix.

Normal (c): cusp singularity. (X, p) where the minimal resolution (X, D) → P1 2 − (X, p)hasforD acycleof ’s Ei with all Ei 2 E 1 E 2

E 3 E4

2 − and the least one Ei 3. It seems plausible, and may in fact be known, that the − 2 Ei are determined by the spectrum of Tss. P1 For the cusp, ResEi (ω) is a 1-form on with log poles at the two interssection points. At a point of Ei ∩ Ei+1 the residues are opposite. Hence for each cusp the pg can drop by at most 1 in the limit. Non-normal (a): X has a smooth double curve C with pinch points (Whitney swallowtail given locally by x2y = z2).

Non-normal (b): Informally stated, X has a nodal double curve with pinch points whose minimal resolution has a cycle of P1’s. These surfaces are frequently constructed by a gluing construction that will be illustrated below.

Example of non-normal (a): Let C ⊂ P2 be a smooth plane quartic having an involution τ : C → C with quotient D = C/τ an elliptic curve. Then X = P2/τ is a surface having a smooth double curve D with pinch points at the 4 branch points of C → D. The desingularization X of X is P2 and by pulling back 2-forms one has that 0 ∼ 0 2 − ∼ 0 − H (KX ) = H (Ω (log C)) = H (OP2 (1)) are the τ anti-invariant 2-forms on P2 having a log pole on C.Thus 0 h (KX )=2 2 KX =1 so that X “looks like” an I-surface. In fact X canbesmoothedtosuch([FPR15a, FPR15b, FPR17]).

non-normal (b) ([LR16]): This is a degeneration of the preceeding example. Before explaining it we will give a general contextual comment.

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Conjecturally an I-surface analogue of the “most degenerate,” meaning no eq- uisingular deformations, genus 2 curve

or $ (dollar bill curve)

obtained by identifying three distinct points on each of two P1’s is the surface obtained by identifying pairs of lines in a quadrilateral

L2 L3 P2

L4 L1

Here to obtain a well-defined involution⎧ τ of⎫ the quartic curve given by the quadri- ⎨12 ←→ 21⎬ ←→ lateral we identify L1 and L2 by ⎩13 24⎭ and similarly for L3 and L4.Inmore 14 ←→ 23 detail, to identify two P1’s we need to say how three points on each are identified. Setting ij = Li ∩ Lj the identification of L1 with L2 is described by the procedure in the brackets. The construction is illustrated by the picture

Here the dotted lines are the blowups of the intersection points of the original four lines. On this blowup P2 the involution τ is well deﬁned, and the cycles of P1’s are obtained from those in the picture. We will return to this example later. B. Hodge theory: Traditionally there have been two principal ways in which Hodge theory interacts with algebraic geometry: • topology; as previously noted many of the deeper aspects of the topology of an algebraic variety X areprovedviaHodgetheory;26

26This was true initially when X is smooth. Using mixed Hodge theory it is now the case when X is arbitrary (singular, non-complete or both), and as will be discussed below it is also the case when we have a degeneration Xt → X leading for example to a proof of the above monodromy n theorem and deﬁnition of the deﬁnition of limt→0 H (Xt). There is also a very rich and beautiful Hodge theory associated to isolated hypersurface singularities.

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• geometry; the Hodge structure on cohomology and its 1st order variations have been used to study the geometry of an algebraic variety X, especially the algebraic cycles that lie in X and in varieties constructed from X. A central point of these notes is to possibly add a third point to this list: • it is now well understood how Hodge structures can degenerate to a limiting mixed Hodge structure; this can then be used to guide and complement the study of algebraic varieties acquiring singularities as occur in moduli. What is meant by a Hodge structure (HS), a mixed Hodge structure (MHS) and a limiting mixed Hodge structure (LMHS)? Traditionally a HS or a MHS was given by a period matrix ωi Γα

where the ωi are rational (meromorphic) diﬀerential forms on an algebraic variety 27 X and the Γα are cycles (including relative ones). When X is smooth and the ωi are regular (holomorphic) n-forms this gives a holomorphic part 0 n n,0 ⊂ n ∼ n C H (ΩX )=H (X) HDR(X) = H (X, ) of the cohomology of X. As noted above, when n =dimX =1, 2 using conjugation 0 n and the cup product in cohomology the holomorphic part H (ΩX ) determines the Hodge decomposition ⎧ ⎨Hn(X, C)= ⊕ Hp,q(X) p+q=n ⎩ Hp,q(X)=Hq,p(X)

on cohomology, where using the isomorphism given by de Rham’s theorem cohomology classes represented by C∞ Hp,q(X)= diﬀerential forms of type (p, q)

One deﬁnes a Hodge structure of weight n (V,F•) to be given by a Q-vector space V and a decreasing Hodge ﬁltration

n n−1 0 F ⊂ F ⊂···⊂F = VC

that satisﬁes − p n p+1 ∼ F ⊕ F −→ VC

We note that much of the use of Hodge theory in birational geometry centers around the implications (primarily to vanishing theorems) of the surjectivity of the natural map H1(X, C) 1 H (OX ). There is a natural splitting of this map. Also, of course the mere existence of functorial mixed Hodge structures and the strictness property of morphisms between them is of frequent use (cf. [KK10]). The actual geometry arising from Hodge structures together with their extensions and 1st order variations has thus far not played a particularly significant role. 27Classically (Riemann, Picard, Lefschetez,. . . ) there were differentials of 1st 2nd and 3rd kinds. It is now understood that the first kind deals with the holomorphic part of the Hodge theory of smooth varieties, the second kind with the full cohomology of smooth varieties and the third with the mixed Hodge theory of singular varieties.

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for 0 p n.28 The relations ⎧ ⎨F p = ⊕ V p ,q pp ⎩ q V p,q = F p ∩ F give a 1-1 correspondence between Hodge filtrations and Hodge decompositions ⎧ ⎨ p,q VC = ⊕ V p+q=n ⎩ p,q V = V q,p. The reason for using Hodge filtrations is that the F p(X) vary holomorphically with X. In practice there will also be a lattice VZ ⊂ V that represents integral cohomology. When X is of dimension n the cup products on cohomology and relations extending ω ∧ ω = 0 (because ω ∧ ω =0) X ∧ ∧ 29 cn X ω ω>0 (because cnω ω>0) for holomorphic n-forms lead to the definition of a polarized Hodge structure (V,Q,F•) where Q : V ⊗ V → Q,Q(u, v)=(−1)nQ(v, u) and the following two Hodge-Riemann bilinear relations are satisfied: (I) Q(F p,Fn−p+1)=0; p,q (II) ip−qQ(V p,q, V ) > 0. • A mixed Hodge structure is given by (V,W•,F ) where the increasing weight filtration W0 ⊂ W1 ⊂···⊂Wm is defined over Q, and where the Hodge filtration F • induces on the graded quotients W Grn V = Wn(V )/Wn−1(V ) p W p ∩ a Hodge structure of weight n.HereF Grn V = F Wn(V )/Wn−1(V ). Mixed Hodge structures have wonderful linear algebra properties. They form an abelian category,30 and any morphism ϕ :(V,W,F) → (V ,W,V)isstrict in ∩ ∩ p p the sense that ϕ(V ) Wk = ϕ(Wk)andϕ(V ) F = ϕ(F ). The basic results connecting Hodge theory to the cohomology of algebraic varieties are • for X smooth and complete, Hn(X, Q) has a Hodge structure of weight n (Hodge). As noted above, for m = n = 1 the HS is determined by the period matrix

∈ 0 1 ¨ ωα H (ΩX )(dim=g) ¨ Ω= ωα H ∼ 2g γi H γi ∈ H1(X, Z)(= Z )

28One may think of F p as represented by diﬀerential forms of degree n having in holomorphic local coordinates z1,...,zn at least pdzi’s.

30Polarized Hodge structures constitute a semi-simple category. Although many Hodge structures occurring in geometry have no natural polarization, the proofs of the deeper results in Hodge theory and its applications to topology require the Hodge structures to be polarizable.

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2 0 2 1 2 1⊥ For m = n =2theHSonH (X) is determined by H (ΩX )=F by F = F ; 0 2 as in the curve case H (ΩX ) is given by the period matrix for the holomorphic 2-forms. Thus for both curves and surfaces the PHS is determined by the classical period matrix. For a general complete algebraic variety X, Hm(X, Q) has a mixed Hodge structure where the weight ﬁltration is W0 ⊂···⊂Wm (Deligne).

AK 6 6 A

W0 W1 The picture is meant to suggest that MHS on Hm(X, Q) is constructed from the pure HS’s on the strata of a desingularization of X; this is (very non-trivially using homological algebra constructions — cf. [CMSP17]and[PS08]) indeed the case. The use of Hodge theory to study a degenerating family Xt → X0 = X of algebraic varieties leads to the notion of a very special type of mixed Hodge structures, namely that of a limiting mixed Hodge structure (V,W(N),F•). Here W (N)isthe monodromy weight ﬁltration constructed from the logarithm N of the unipotent part of monodromy. Assuming N n+1 =0,N n = 0, it is the unique ﬁltration

W0(N) ⊂ W1(N) ⊂···⊂W2n(N) satisfying N : Wk(N) → Wk−2(N) k ∼ 31 N : Wn+k(N) −→ Wn−k(N). • Then a LMHS is given by a MHS (V,W(N),Flim)where p → p−1 N : Flim Flim . ∼ 2m The associated graded Gr(LMHS)= ⊕ H where H is a HS of weight ; the picture =0 is a Hodge diamond. Here m =2andN is the vertical arrows; the dots are the Hp,q’s: (2,2)

(1,2) (2,1) (2,0) (0,2) (1,0) (0,1)

(0,0) We will set hp,q = dimension of the (p, q)dot. Theorem (Schmid).32 Given X → Δ as above m lim H (Xt)=LMHS. t→0 The proof is a combination of • Lie theory

32Cf. [Sch73]and[CKS86] in the references. An algebraic approach may be found in in [PS08].

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• complex analysis • diﬀerential geometry In simplest terms, the period matrices of a degenerating family of varieties have entries that are polynomials in log t and with holomorphic functions as coeﬃcients. Monodromy is given by analytic continuation of log t around t =0.Theabove ingredients, especially Lie theory, are then used to give a precise description of the asymptotic behavior of the period matrix. The following is a typical picture one has in mind of a family of varieties Xt parametrized by the disc Δ and which are smooth for t ∈ Δ∗ = {t ∈ Δ:t =0 } while X0 has acquired singularities.

X∗ ⊂ X

Δ∗ ⊂ Δ

to The following picture represents a family of genus 2 curves acquiring a node:

Xt X0

Associated to a general family of degenerating smooth varieties is a monodromy n n operator T : H (Xt) → H (Xt) T = TssTu (Jordan decomposition) k N n+1 Tss = I,Tu = e with N =0. It is the basic topological invariant associated to the family X∗ → Δ∗. For the above degeneration of a genus 2 curve the LMHS is

(1, 1) s ss (1, 0) (0, 1) s (0, 0)

The solid vertical line represents the action of N on the associated graded to the LMHS (cf. Chapter 4 in [CMSP17]).

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Referring to the picture of M2 in the introduction, the solid lines in the diagram represent degenerations with N =0. The following illustrates the type of pictures one has in mind for a LMHS:

• - topological picture

2 • - y = x(x − 1)(x − t) algebraic picture

• X = C/Λ, Λ = {1,λ}

λ analytic or Hodge-theoretic picture

1 → aλ+b ab ∈ Z M ∼ Here λ is determined up to λ cλ+d where cd SL2( )and 1 = SL(2, Z)\H, H = {λ :Imλ>0} ∗ 01 1 λ ∈ P1 In this case V =(∗ ), Q = −10 , F =[1 ] , HR II corresponds to Im λ>0 H ⊂ P1 11 and the space of PHS’s is . The monodromy T =(01) is translation in the → ∞ 1 → 1 1 33 upper half plane, and as λ i we have F [ 0 ]=Flim.

log t λ = t 2πi

∞ 6

λt

How does Lie theory enter? The actors in the story are • n 0 • • Period domain D = {F =ﬂag F ⊂ ··· ⊂ F = VC} in VC :(V,Q,F )= PHS ; • compact dual Dˇ = F • is a ﬂag with Q(F p,Fn−p+1)=0 ; • G =Aut(V,Q)isaQ-algebraic group;

33 This picture is not indicative of what happens in the non-classical case. The Flim will not in general lie in the boundary of the period domain in its compact dual (see below).

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• GR acts transitively on D and GC acts transitively on Dˇ so that we have

D = GR/H with H compact ∩ ∗∗∗∗ ◦∗∗∗ Dˇ = GC/P with P parabolic ◦◦∗∗ ◦◦◦∗

where D is an open GR-orbit in Dˇ. Examples: 1,0 • m =1: D =Sp(2g, R)/U(g)=Hg where g = h ; • m =2: D =SO(2k, )/U(k) × SO()wherek = h2,0, = h1,1. Theclassicalcaseiswhenwehave D = Hermitian symmetric domain (HSD) =

GR/K, K = maximal compact. In algebraic geometry these arise as m = 1 (curves, abelian varieties); m = 2 is HSD ⇐⇒ k = 1 (K3’s).34 Thus h2,0 2 is non-classical. For n 3andX Calabi-Yau, the D corresponding to Hn(X) is also non-classical. Period domains have sub-domains corresponding to PHS’s with additional structure; e.g., D ⊂ D

= reducible PHS’s that are ⊕’s

This is what the dotted lines represent in the diagram in the Section I.A for M2.In general one has Mumford-Tate sub-domains of D, deﬁned to be those PHS’s with a given algebra of Hodge tensors. Period mappings arise from holomorphic mappings ⎧ ⎫ ⎨equivalence⎬ → \ Φ:B ⎩ classes of ⎭ =Γ D PHS’s

where B is a complex manifold and Γ ⊂ GZ contains the monodromy group. One may think of B as the parameter space for a family of smooth algebraic varieties ∈ n Xb, b B, whose cohomology groups can be identiﬁed with H (Xb0 ) for a base ∈ n point b0 B up to the action by monodromy of π1(B,b0)onH (Xb0 ). Example: As noted above the ﬁrst non-classical case is weight n =2whenh2,0 =2. In this case D has an invariant contact structure and the image of any period mapping Φ is an integral variety of that structure. This means that if the contact structure is given by a 1-form θ, which up to scaling is invariant by GR,then Φ∗(θ)=Φ∗(dθ)=0.

34 In this case D is a type IV HSD; it may be equivariantly embedded on Hg. Also the unit ball may be equivariantly in Hg. Thus the classical case should probably be deﬁned as referring to algebraic varieties whose PHS’s lie in a Mumford-Tate sub-domain of Hg.

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In general the diﬀerential constraint satisﬁed by period mappings in the non- classical case is the basic new phenomenon that occurs. Thus Φ(M)cannotcontain an open set in D, Γ need not be arithmetic, etc.

Note: For families of algebraic surfaces with pg = 2 we think of the period mapping Φ as given by a holomorphically varying 2 × k matrix Ω satisfying HRI expressed by ΩQtΩ = 0. Differentiating this relation gives that the 2 × 2matrixdΩQtΩ= −t(dΩQtΩ) is skew symmetric; writing 0 θ dΩQtΩ= −θ 0 the 1-form θ gives the contact structure. Using Lie theory the set of equivalence classes of LMHS’s has been classified [Ker15]; they form a stratified object. As noted above one may informally say that we know how Hodge structures degenerate; the strategy is then to use this information to help understand how algebraic varieties degenerate. Examples: For n = 1 the stratification may be pictured as

I0 I1 ··· Ig . reﬂecting (for g =2)

. . .

For n =2withh2,0 = 1 the picture is I II III with corresponding Hodge diamonds

1 b 1 I rrr

11 rr b − 2 II r N 2 =0,rankN =2 ? ? rr

1r ? r 2 2 III b N =0,rankN =rankN =1 ? r

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In the case of moduli of K3’s, case I corresponds to smooth polarized K3 surfaces.35 In case II, Hodge theory suggests is that the surface acquires an elliptic double curve. Case III is the case when the LMHS is of Hodge-Tate type. Dating from the work of Kulikov in the 70’s, the moduli of K3’s is a much studied and very beautiful story (cf. [Fri84]and[Laz16]). For n = 2 the picture is

{ II FF {{ FFF {{ 0 IIVC V CCC xx C xxx III The detailed picture with the ranks of N and N 2 filledinwhenh2,0 = 2 will be given below. In all of these examples the Roman numerals reflect the associated graded to the equivalence classes of LMHS’s. The stratification is linear and transitive in the classical case, transitive but not linear in the non-classical n =2case,and neither transitive nor linear in the general n 3case. Within each of the above strata there is a refined stratification given by PHS’s with “additional” Hodge tensors (Mumford-Tate sub-domains). We emphasize that the Hodge-theoretic stratification has the following ingredients: (a) the equivalence classes of LMHS’s over Q; (b) the spectra (basically the eigenvalues) of the semi-simple parts of monodromy; (c) for each equivalence class of LMHS’s over Q, within the PHS’s given by the associated graded there is a stratification by Mumford-Tate sub-domains.

For the I-surface example we will see that the part of the stratiﬁcation of MI that has been determined is faithfully captured by using all of the Hodge-theoretic ingredients (i), (ii), (iii) above. Example: Curves with N =0. Using the proper Mumford-Tate sub-domain given by PHS’s that are non-trivial direct sums over Z one may Hodge-theoretically detect the degeneration

which has trivial monodromy. In general, in the stratification of M2 pictured in the first section of these notes the solid lines refer to degenerations where N =0and the dotted lines to degenerations where the Jacobian of the normalization splits further into a direct sum of principally polarized abelian varieties. Example: n =2. At least in some examples one may Hodge theoretically detect 1 a degeneration to a d (1,a) singularity where N = 0. The first case is the Wahl 1 singularity 4 (1, 1) where T =Id;thenforI-surfaces X with one such singularity there is an outline of an argument that the image in the period domain picks up an extra Hodge class. It is known that having this singularity defines a non-empty boundary divisor in moduli, and then the tentative result is that where the image

35Suitably interpreted it also includes the case where X is nodal.

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of the period mapping meets the Mumford-Tate domain corresponding to the extra Hodge class will deﬁne that boundary divisor. This will be further discussed below.

III.A. Generalities on Hodge theory and moduli. The ﬁrst point is that there is a moduli space H =Γ\D

for Γ-equivalence classes of PHS’s (think of D = H and Γ ⊂ SL2(Z)). Thesecondpointisthatthereisaperiod mapping (∗)Φ:M → P ⊂ Γ\D where Γ contains the global monodromy group given by the image of the monodromy representation

ρ : π1(M) → GZ. There are several important technicalities here, some dealing with the singularities of M and some with the presence of X’s with extra automorphisms. And of course the general X may not be smooth but rather will have canonical singularities. In the example of the I-surface to be discussed next these issues can be addressed directly. The following are statements that have been proved at the set-theoretic level and full results established under various assumptions; complete proofs are a work in progress (cf. [[Gri69]] for a discussion of this). Theorem A. The image P =Φ(M) ⊂ Γ\D is a quasi-projective variety that has a canonical projective completion P.36 Set-theoretically, P is obtained from P by attaching the associated graded to the limiting mixed Hodge structures arising from Φ in (∗). We shall call P the Satake-Baily-Borel (SBB) completion of P. In the classical case using the Borel extension theorem the P is induced from the classical Satake- Baily-Borel compactiﬁcation of arithmetic quotients of HSD’s. In general P is the minimal natural Hodge theoretic completion of P.One may think of it as throwing out the extension data in the LMHS. For Γ arithmetic and with the assumption of the existence of a fan, Kato-Usui in [KU09]havecon- structed a universal maximal completion of Γ\D, one in which the extension data is included. It may be thought of as a Hodge-theoretic toroidal compactiﬁcation of Γ\D. Note: The proof of the above theorem (if completed) will have the following algebro-geometric implication: Let f Y −→ Z be a morphism of smooth, projective varieties and assume that the relative dualizing sheaf ωY/Z is a line bundle. Then

(∗∗)Λ=:detf∗(ωZ/Y )issemi-ample. It is known that Λ is nef, and if local Torelli holds for a general point of Z,thenΛis big [[Gri69]]. The freeness seems more subtle (witness the abundance conjecture).

36Cf. [BBT] and [BK] for an interesting “model-theoretic” proof of the result that when Γ is arithmetic P is quasi-projective.

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One may ask: Once you know that Λ is big and nef, why don’t the standard methods of birational geometry (the minimal-model-program, including the base- point-free theorem) apply to give a proof? The interesting answer is that the signs needed in the base-point-free theorem are opposite to those that occur in the above situation. The connection of this statement with the above theorem is that if X −→f M is a versal family of general type varieties, then P =Proj(Λ). Thus assuming (∗∗) one may deﬁne the SBB completion of the image of the period mapping without using any Hodge theory.37 The second work-in-progress result is Theorem B. The period mapping Φ extends to

(∗∗)Φe : M → P. The above two structural statements provide a conceptual framework for the use of Hodge theory to partner with and help guide the standard algebro-geometric methods used to study the boundary structure for the KSBA moduli spaces for surfaces of general type.38 How this works will now be illustrated.

III.B. I-surfaces and their period mappings. Murphy’s law (Vakil): Whatever nasty property a scheme can have already occurs for the moduli spaces of general type surfaces. Thus unlike curves one should select “special” surfaces to study. In geometry extremal cases are frequently interesting; Noether’s inequality K2 p (X) X +2 g 2 suggests studying surfaces close to extremal. The 1st non-classical case is given by the Deﬁnition: An I-surface X is a regular (q(X) = 0) general type surface that satisﬁes 2 pg(X)=2,KX =1. Here we are assuming that X is either smooth or has canonical (du Val) singularities. The KSBA moduli space for these surfaces will be denoted by MI . One studies general type surfaces via their pluri-canonical maps ∗ − P 0 ∼ PPm 1 () ϕmKX : X H (mKX ) = 0 and pluricanonical rings R(X)=⊕H (mKX ). Instead of () it is frequently better to use weighted projective spaces corresponding to when we add new generators to R(X). From

Pm(X)=m(m − 1)/2+3,m 2

37 Of course even if (∗∗) is proved algebro-geometrically, just taking the Proj of det(f∗ωX/Y ) does not seem to even roughly suggest what singularities are added on the boundary. 38It is not necessary to have proofs in order to use the statements of these results to help to guide how one may understand moduli.

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and Kodaira-Kawamata-Viehweg vanishing (cf. [Dem12]) one has for the I-surface P1 | | ϕKX : X , KX = pencil of hyperelliptic curves → P → P3 ϕ2KX : X (1, 1, 2) of degree 2; → P → P12 ϕ5KX : X (1, 1, 2, 5) an embedding.

Here the homogenous coordinates of the mappings ϕmKX are given by the generators of R(X) in the indicated degrees. If C ∈|KX | is a general smooth ﬁbre, then by adjunction 2KX C = KC .

Thus the images ϕ2KX (C)=ϕKC (C) are canonical curves. The I-surface was

important classically since ϕ4KX is not birational, while for any general type surface

ϕ5KX always is birational. For the⎧ I-surface we have the following equations and picture: ⎪ ⎪ ⎪ ⎨ P(1, 1, 2) → P3given by • P 2 2 ⎪ (t0,t1) → [t ,t0t1,t ,y] ⎪ V 0 1 ⎪ ⎩⎪ X = 2:1 map branched over P and V ∈|OP3 (5)| • The canonical pencil |KX | has base point P and is given by the two sheeted coverings of the lines L through the vertex in the quadric and branched over P and L ∩ V . 2 • The equation of X is z = F5(t0,t1,y)z + F10(t0,t1,y) (weighted hypersurface in P(1, 1, 2, 5)). One may extend the deﬁnition to include surfaces X whose canonical Weil divisor class KX is Q-Cartier and which have semi-log-canonical singularities and which have the numerical properties of smooth I-surfaces (cf. [Kol13a]). Example: At the other extreme to the smooth I-surfaces is the surface

discussed above and which is obtained from (P2, 4 lines) by identifying opposite pairs of lines. For this surface the equation is (cf. [LR16]) 2 2 − 2 2 − 2 z = y(t0 y) (t1 y) . Geometrically it is a double cover of a quadric cone in P3 branched over the vertex, a plane section, and two double plane sections. A general curve in |KX | is

Concerning the moduli space MI for smooth I-surfaces we have

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• MI is smooth and 1 – dim MI = h (TX ) = 28, – dim DI =57=2dimMX +1, • Φ:MI → ΓI \DI has injective diﬀerential Φ∗ (local Torelli) and this implies that

• Φ(MI ) is a contact submanifold P → ΓI \DI , • ΓI is arithmetic; it is not known is whether Γ = GZ or not. Elaborating a bit on a previous statement, in general, in the non-classical case there is a non-trivial homogeneous sub-bundle E ⊂ TD such that any period mapping satisﬁes 39 Φ∗ : T M → E ⊂ T (Γ\D); thus the image can never be an open set in Γ\D. Moreover, although it is always thecasethat vol Φ(M) < ∞,

ItcanhappenthatΓ⊂ GZ is a thin subgroup, i.e., a subgroup with [Γ : GZ]=∞. Stratiﬁcation of the space of Gr(LMHS)’s: For curves with Γ = Sp(2g, Z)we have for LMHS’s _ _ ··· I0O I1O I2O IgO

Hg Hg−1 Hg−2 H0. 2 Note that Ig−m corresponds to N :Gr2 → Gr0 with N =0,rankN = m. m t Gr2 g − m tt ∼ 1 Gr1 (= H (C)) ? t Gr0 For each boundary component we have the stratiﬁcation 1 ⊕ 1 H = Hi .

The composite of these induces a stratiﬁcation of Mg by {# nodes, # components}.

Of course this is just the beginning of the story of Mg. For K3 surfaces there is an extensive literature dealing with the stratification of the image of the period mapping extended to include singular K3’s for which N = 0 (Heegard divisor, etc.); cf. [Laz16] for an analysis of degree 2 K3’s and further references. The stratification diagram for the possible LMHS’s was given above. 2 For surfaces with pg = 2 a refinement of the earlier picture by N =0, N =0 of the classification of Gr(LMHS)’s/Q is

39E is deﬁned by the diﬀerential constraint • − F p ⊆ F p 1.

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2 0 N = 0 1 2 I 1 N = 0, rank N = 2

1 2 2 N = 0, N = 0, rank N = 1 II III rank N = 4

1 1 N 2 = 0, rank N=3 and IV rank N2 = 1

2 N 2 = 0, 2 V rank N = 2

For the reﬁned Hodge-theoretic stratiﬁcation of Gr(LHHS/Z)’s we use Tss → {conjugacy class [Tss]ofTs in Γ}. Within each of these strata we use Mumford-Tate sub-domains appearing in Gr(LMHS)’s in MI . MGor ⊂ M We begin by considering the Gorenstein part I I . One reason for this is the general result

if Xt → X is a KSBA degeneration of a surface where all the singularities of X are isolated and non-Gorenstein, then N =0. Hence only Gorenstein singularities can non-trivially contribute to the LMHS/Q.40 Heuristically the reason for this is the following. • For the resolution of the singularity of a non-Gorenstein slc singularity 1 one has a divisor D = Ei where the Ei are P ’s and the dual graph is a chain or perhaps a Dynkin-like diagram with forks; there are no cycles. • For a KSBA degeneration Xt → X with X → X a desingularization, 0 2 0 2 and ωt ∈ H (Ω ), the limit limt→0 ωt = ω ∈ H (Ω (log D)) and then Xt X ResD ω gives a meromorphic 1-form on the Ei’s with log poles on Ei ∩ Ej and thus ResD ω = 0. It follows that pg(X)=pg(X), which then implies that N =0.41 The following results from coupling the classification in [FPR15a, FPR15b, FPR17] with the analysis of the LMHS’s in the various cases. Theorem B. The Hodge theoretic stratification of P given by the above diagram MGor via the extended period mapping uniquely determines the stratification of I . Moreover, any Hodge-theoretic degenerations that are possible as pictured in the table below are actually realized by Gorenstein degenerations of I-surfaces.

40We recall that for a normal surface X, Gorenstein means that the canonical Weil divisor class KX is a line bundle. In general the index is the least integer m such that mKX is a line 1 bundle. For the 4 (1, 1) singularity the index is 2. A central general question in moduli is to determine a useful bound on the index. 41This is a consequence of the Clemens-Schmid exact sequence; cf. [CMSP17]and[PS08].

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Rather than display the whole table the following is just the part for simple 2 elliptic singularities (types Ik and IIIk). They have N =0sinceforthesemi- stable-reduction (SSR) of a degeneration only double curves (and no triple points) occur; all of the other types occur if we include cusp singularities. In the following, • 2 X is irreducible (since KX is a line bundle with KX = 1 and any compo- 2 nent of X will have positive KX ) • di = degree of elliptic singularity • k = # elliptic singularities — in general, as previously noted using Hodge theory one may show that k pg +1 • X = minimal desingularization of X —inaSSRgivenbyX → Δthe surface X will appear as one component of the ﬁbre over the origin. In the following table, in the 1st column subscripts denote the degrees of the 42 elliptic singularities, which one can show are uniquely determined by the [Tss]’s. We will explain the (9 − di) column below in the appendix.

k minimal codim stratum dimension (9 − di) k resolution X in MI i=1

I0 28 canonical singularities 0 0 0 blow up of I2 20 a K3-surface 718

minimal elliptic surface I1 19 with χ(X)=2 819

III2,2 12 rational surface 14 2 16

III1,2 11 rational surface 15 2 17

III1,1,R 10 rational surface 16 2 18 blow up of an III1,1,E 10 Enriques surface 16 2 18 ruled surface with III1,1,2 2 χ(X)=0 23 3 26 ruled surface with III1,1,1 1 χ(X)=0 24 3 27 Note that the last column is the sum of the two columns preceding it. Appendix: We will will use Hodge theory as a guide to study the boundary component MGor ⊂ M 2 I whose general point is a normal I-surface X that arises from a KSBA degeneration (∗) X → Δ whose monodromy logarithm N satisﬁes N 2 =0 rank N =2

42 The Tss contains the Coxeter element of the Dynkin diagram. By inspecting the table of Coxeter elements for simple elliptic singularities with indices 1 and 2 one may check this statement.

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and where the degree of the specialization (∗) is 2 (we will explain below how to Gor deﬁne the degree of the specialization). As above, M denotes the part of M parametrizing surfaces with only Gorenstein singularities. With these assumptions we will see that X is irreducible with a single normal singular point p. Hodge theory then gives that p is a simple elliptic singularity whose degree is in this case deﬁned to be degree of the specialization. In this situation one has the following diagram: ∗∗ ( )(X,C<) || << g ||

Φ:MI → P ⊂ Γ\D. As mentioned above, using other methods one may also prove that local Torelli holds everywhere for this period mapping; a result that was independently obtained by Carlson-Toledo and by Pearlstein-Zhang. It seems quite possible that the methods

43 k − This is the “7” that appears in the i=1(9 di) column in the above table.

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of Friedman (cf. [Fri84]) may be used to prove generic global Torelli, meaning that the period mapping Φ : MI → P has degree one. Conclusion: Using Hodge theory as a guide one may determine the structure of the boundary component M2 of MI as well as a desingularization of MI along M2. Using the results of [FPR15a, FPR15b, FPR17], together with them we are MGor working to extend the above conclusion to the other boundary components of I as listed in the above table. We will also discuss the non-Gorenstein components below. Step one: The Hodge diamond for the LMHS associated to a semi-stable reduction X → Δ arising from a KSBA family with X as central ﬁbre in 11rr 11rrr27 ? ? rr where the dimensions are written in above the dots. This suggests (does not prove) that the central ﬁbre X0 of X → Δ is a normal crossing surface of the form ∪ X0 = X C Y where • X is a desingularization of X; • C is a smooth double curve. In this situation the LMHS is computed from the groups Hp(C), Hq(X) using Gysin and restriction mappings (cf. [PS08]). Assuming that H1(X)=0andH1(Y )=0 this gives • H1 :=Gr1(LMHS) = H1(C); • H2 :=Gr2(LMHS) is the middle cohomology of the complex Gy () H0(C)(−1) −−→ H2(X) ⊕ H2(Y ) −Re−→ H2(C) where Gy is the direct sum of Gysin maps and Re is the signed restriction map. ∪ Note: The condition [Fri83]thatX C Y be smoothable is ∼ ∗ () N N . C/X = C/Y Denoting by CX the curve C considered in X and similarly for CY ,()gives C2 + C2 =0, X Y which is exactly the condition that Re ◦ Gy = 0 in ()above. Step two: C is a smooth elliptic curve and therefore can be realized as a smooth P2 − cubic in .Inordertohave deg NC/ X =degNC/Y = 2 in ()toobtainY we 2 must blow up P in 7 points pi and take the proper transform of the cubic curve to obtain ⊂ P2 44 C Y =Bl{pi} .

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There is still one parameter to adjust to have () as an equality of line bundles. From dimensions in the Hodge diamond and dim H2(Y )=8weobtain dim H2,0(X)=1 dim H1,1(X)=21.

Now we use intersection numbers. From the assumption that (X, C) → (X, p) is a minimal resolution of an I-surface and adjunction we obtain 2 (KX + C) =1 · 2 ⇒ · KX C + C =0 = KX C =2.

2 2,0 From this we infer that K = −1, and since h (X) = 1 the line bundle K is [E] X X for E a(−1)-curve in X. This gives the diagram (∗∗) which may be pictured as

C E

X p C

Xmin We then have ∼ • O KXmin = Xmin (since KX =[E]); 1,1 • h (Xmin) = 20; 2 • C =2 =⇒ pa(C)=2,g(C)=1. Step three: We now observe that this construction is reversible. Given a K3 surface Xmin with a degree 2 polarization and a curve C with pa(C)=2,g(C)=1wehave the situation described in (iii) above. We then construct X by blowing up the node on C to obtain C with C2 = −2. Contracting it then gives (X, p). We next count parameters. First we have dim Xmin’s = 19 dim C’s = 1

44In [Fri84] type II degenerations are constructed by ﬁrst gluing two P2’s together along a common cubic curve C. To obtain the condition () one has to blow up 18 points on C.Asshown in [Fri84] there are exactly four ways to group these points into two groups that specify in which P2 the blowups occur. It may be that in the situation being discussed here there is a unique such choice.

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so that the boundary component M2 has dimension 20. This gives the dimension count in (ii). ∪ For the dimension count for the semi-stable surfaces X C Y with a fixed (X, p) we have ⎧ 2 ⎨dimension of the spaces of pi’s in P =14 + ⎩dimension of cubics through the pi =2 16 and ⎧ ⎨ parameter to have ()1 − ⎩ dimension of Aut(P2) 8 9 This gives dimension of the surfaces X ∪ Y C =16− 9=7=28− 1. with a fixed (Xmin,C) ∪ Letting (Xmin,C) now vary one may show that the space of X C Y ’s forms a smooth variety of dimension 27 biregularly equivalent to the blowup of MI along the 20 dimensional M2. Step four: Finally how does Hodge theory enter via Torelli to relate to the above parameter counts? With the details to appear in a later work the rough idea is • H1 determines the elliptic curve C; • H2 = H2 ⊕ H2 where H2 istheHodgestructureofaK3withadegree 2 polarization, and H2 is a lattice of rank in Hg(H2)ofrank7; • 1 2 1 the extension data in ExtMHS(H ,H ) gives the set of 7 points pi on C. There are a number of subtleties involving the quadratic forms on weight 2 PHS’s which are a work in progress. We refer especially to [Fri84] where similar constructions are carried out in the case of K3’s, and where a first instance of the use of extension data to provide parameters for boundary components in moduli of surfaces appeared. In summary, Hodge theory suggests where to look — the seven parameters arise from the possible extension data for GR(LMHS) — and following [FPR15a,FPR15b,FPR17] in the references), and on discusssions that the four of us have had with them related to a joint project that is in progress.

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IMSA (Institute of the Mathematical Sciences of the Americas) at the University of Miami, and the Institute for Advanced Study Email address: [email protected]

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